Theorem fusgrvtxfi 29406

Description: A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.)

Hypothesis

Ref	Expression
isfusgr.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
fusgrvtxfi	⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Proof of Theorem fusgrvtxfi

Step	Hyp	Ref	Expression
1		isfusgr.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	isfusgr 29405	. 2 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3	2	simprbi 498	1 ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  Fincfn 8883  Vtxcvtx 29083  USGraphcusgr 29236  FinUSGraphcfusgr 29403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-fusgr 29404

This theorem is referenced by:  fusgrfupgrfs  29418  nbfusgrlevtxm1  29464  nbfusgrlevtxm2  29465  nbusgrvtxm1  29466  uvtxnm1nbgr  29491  cusgrm1rusgr  29669  wlksnfi  29993  fusgrhashclwwlkn  30167  clwwlkndivn  30168  fusgreghash2wsp  30426  numclwwlk3lem2  30472  numclwwlk4  30474